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Greenland sea bottom water formation: a balance between advection and double-diffusion
Deep-SeaResearch.Vol.30, No. I IA. pp. 1109to I 117, 1983. Printed in Great Britain.
0198-0149/83 $3.00 + 0.00 O 1983Pergamon Press Ltd.
Greenland Sea Bottom Water formation: a balance between adveetion and double-diffusion TREVORJ. McDOUGALL* (Received 31 January 1983; accepted 5 April 1983; final revision received 19April 1983) Abstract--Greenland Sea Bottom Water is believed to form by the subsurfzce modification of Atlantic Water (CARMACK and AAOAARD, 1973, Deep-Sea Research, 20, 687-715) rather than by direct cooling at the surface. Both double-diffusive convection and cabbeling have been suggested as responsible for the increase in density of the Atlantic Water, but we argue against the importance of cabbeling. A simple model is presented in which there is a balance between advection and double diffusion; it is applied to the Atlantic Water as it flows horizontally towards the centre of the Greenland Gyre. The model gives a prediction for the rate of formation of Bottom Water that is encouragingly close to that observed.
INTRODUCTION
CARMACK and AAGAARD(1973) conducted a volumetric census of water masses in the Greenland Sea for both summer and winter and found that the volume of Bottom Water (temperature < - I ° C ) varied seasonally by (30 + 10) x 103 km 3. This is a lower limit to the annual production of Bottom Water in the Greenland Sea because the actual production rate must also include the volume of Bottom Water that flows out of the Greenland Sea from one summer to the following winter. The budget of dissolved oxygen rules out the surface mixed layer as the major source of Bottom Water and suggests that Atlantic Water, which flows into the Greenland Sea just below the surface layer, is first modified by double-diffusive convection and then sinks as Bottom Water near the centre of the gyre. Figure 1 (from CARMACKand AAOAARD, 1973) shows the temperatures and salinities of both surface water and the subsurface, modified Atlantic Water at several locations in the Greenland Sea. The surface measurements are joined by straight lines to the corresponding subsurface points at the same location. As the centre of the gyre is approached, the salinity of the surface layer increases and the temperature and salinity of the modified Atlantic Water decrease. The temperature of the surface layer remains almost constant at the freezing point. Figure 2 (from CARMACKand AAGAARD, 1973) shows the inflowing Atlantic Water (point A) and the Bottom Water (point BW) on a temperature-salinity (T-S) diagram. Double-diffusive convection (in the diffusive sense) between the well-mixed surface layer and the Atlantic Water causes the temperatures and salinities of the water masses to move in the directions shown by the arrows in Fig. 2. The arrows have been drawn parallel to the line joining points A and BW, and the slope of the line corresponds to a buoyancy flux ratio R/of0.56 Itaking ct=0.5 × 10-4 k -I and 13=0.765 × 10-3 (%0)-I from MAMAYEV, 19751. The density * Research School of Earth Sciences, Australian National University, P.O. Box 4, Canberra 2600, Australia. Present address: Division of Oceanography, C.S.LR.O., P.O. Box 1538, Hobart, Tasmania 7001, Australia. 1109
1110
TREVOR J. McDOUGALL
2f~
...................................................................
I / /£ SA
' 3
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3400
34.20
34.40
34.60
34.80
35.00
35.20
SaLinity (xl03)
Fig. 1. Temperature-pl_init¥ diagram showing surface observations (O)joined to the corresponding points (O) of the modified Atilnti¢ Water layer. The line SA is tangent at the mean temperature - salinity point of Bottom Water BW, to the isopyenal passing.through the point. [From CARMACX and AAOAARD (1973), Fig. 5.]
anomaly ratio, Rp is 1.6 for the two water masses and so while the buoyancy flux ratio is consistent with the measurements by TURNER (1965), it is a little higher than those of
McDoUOALL (198 lb). The line in Fig. 2 connecting the surface water S to the Atlantic Water A is tangent to the isopycnal through point A. This is the conventional criterion for the onset of cabbeling where mixtures of two water masses are more dense than either parent water mass. When two water masses lie one above the other and the criterion for cabbeling is fulfilled, the vertical distributions of temperature and salinity are also in the correct sense for the occurrence of doublediffusive convection. McDOUOALL (1981b) investigated the relative importance of the two
~) 2
E o S'
-I
34.40
34.60
BW
34.80
3500
35.20
Satinity ( x l 0 3)
Fig. 2. Temperature-salinity diagram showing the original (i.e., at the edge of the gyre) Atlantic Water A. the original Surface Polar Water S, and the Bottom Wat~ BW. Double-diffusive fluxes of heat and salt alter A in the direction A' and S in the direction S'. The surface layer also experiences cooling from the atmosphere, which is not shown on the figure. [From CARMACK and AAGAARD (1973), Fig. 7.]
Greenland Sea Bottom Water formation
1111
processes in such geometry and showed that the prime effects of the nonlinearity in the equation of state are firstly to drive a greater level of double-diffusive convective activity in the lower layer than in the upper layer, and secondly, to make the lower edge of the interfacial region less gravitationally stable. Both effects cause the horizontal interface between the water masses to migrate slowly upwards as the lower layer entrains fluid from the upper layer. While cabbeling is effective in causing deep convection by mixing between horizontally adjacent water masses (CARMACK, 1979), McDOUOALL(1981b) showed that when such potentially cabbeling water masses lie one above the other, the vertical static stability of the water column inhibits any large-scale sinking due to cabbeling, and double-diffusive convection is the dominant process. In this paper we seek to understand the processes that cause the properties of Atlantic Water to change as it flows towards the centre of the Greenland Gyre. Firstly we consider a simple model, which is a balance between (a) double-diffusive convection across the horizontal interface and (b) the horizontal advection of the modified Atlantic Water. When the fluid reaches the centre of the gyre it is assumed to have reached the density of.the water below it and then to sink as Bottom Water. The assumption has the nature of a consistency relationship and it gives a production rate of Bottom Water close to that observed. The model is then extended to include firstly the entrainment of surface layer fluid into the modified Atlantic Water. This, together with the oxygen measurements by Carmaek and Aagaard, confirm the conclusions of McDOUGALL(198 l b) that such entrainment is small even though the nonlinearity of the equation of state is significant. Finally, we include the entrainment of Bottom Water into the converging Atlantic Water flow and show that this can at most change our estimate of Bottom Water production rate by 12%. THE B A L A N C E B E T W E E N A D V E C T I O N A N D D O U B L E - D I F F U S I V E C O N V E C T I O N
The basic features of the model are illustrated in Fig. 3. Atlantic Water converges at flow rate Q0 to the central sink in the Greenland Gyre, continually losing heat and salt double diffusively to the surface mixed layer. The fluxes change the temperature and salinity of the converging layer and when the centre of the gyre is reached the fluid is assumed to have the same properties as the underlying Bottom Water, whereupon it sinks. This assumption enables the rate of production of bottom water Q0 to be estimated. If the volume flow rate were > Q0, then double-diffusive convection would not be strong enough to change the layer atmosphere
o°'r:'
I"
"
"
AW
AW
BW
t new
m i
BW
BW
T
T+dT
(a)
o.
(b) Fig. 3. The idealizedmodelin plan view(a) and in cross-section(b). In (a), dA is an elementof the horizontal area between the isotherms T and T + dT. In (b) Atlantic Water (AW) enters at a flow rate Q0 between the surface mixed layer and the Bottom Water (BW).
1 112
I'REVORJ. McDOUGALL
properties to those of Bottom Water and so sinking and the production of Bottom Water would not occur. If, on the other hand, the volume flow rate were
(l)
where a is the thermal expansion coefficient of seawater (defined so a > 0), P0 is a reference density, Cp is the specific heat (at constant pressure) of seawater, and b is a dimensional constant (dimensions of velocity) = 0.323 (g~/v)t, where the symbols have their usualmeaning. For water at room temperature b can be taken as 1.9 × 10 -3 m s -1. AT is the temperature difference across the interface and Rp is the density anomaly ratio, ~ A S / a A T . Consider now the conservation of heat for a control volume of the Atlantic Water layer between the isotherms T and T + dT in Fig. 3a. The horizontal area of the control volume is dA and the depth is the depth of the Atlantic Water layer. The steady-state conservation of heat involves only the terms due to advection and double-diffusive convection and is given by QoadT = - a F r d A .
(2)
The negative sign arises because the modified Atlantic Water is cooled by the double diffusion. Neither the depth of the flowing layer nor the velocity of the flow appear separately in (2), but only the product of these variables multiplied by the circumference of the control volume is relevant and is labelled Q0. We note (as did Carmack and Aagaard) from Fig. 1 that the lines on the T - S diagram joining the successive surface water points to the modified Atlantic Water points have a remarkably constant slope that corresponds to a density anomaly ratio R p of 1.6. The temperature of the surface layer is kept virtually constant at the freezing point by exchange of heat with the atmosphere, and the surface salinity increases due to the double-diffusive flux F s from below. The surface layer also undergoes an unknown amount of adveetion and other processes. Without endeavouring to understand all the processes at work on the surface layer, we make use in our model of the observed constancy of both the surface layer temperature T~ and the density anomaly ratio Rp. We 'define a nondimensional temperature scale by TA-T
0= - -
(3)
TA - T s
where TA is the initial temperature of the inflowing Atlantic Water and T is the temperature of the flow as a function of horizontal position. The temperature and salinity of the modified
Greenland Sea Bottom Water formation
1113
Atlantic Water layer proceed from point A in Fig. 2 towards point BW. The nondimensional temperature 0 varies from zero at point A to 1 when (T, S) is at the point on the extension of the line A to BW where T = T s . From Fig. 1, 0 -~ 0.85 when the temperature and salinity of the modified Atlantic Water are those of Bottom Water (BW). Combining (1), (2), and (3), the advective, double-diffusive balance is dO =
bl(~(T A - Ts)]~
QoR20
(1 - 0)] dA.
(4)
The terms in 0 are readily separated to the left-hand side of the equation and we integrate over the total horizontal area A from 0 = 0 at the outside to 0 = 0 c at the centre of the gyre, giving C -
b [ a ( T A - Ts)]~A
a0R
= f ~ ( 1 - 0)-~d0 = 3(1 - 0~)-~ - 3.
(5)
Because the advective, double-diffusive balance given by (2) or (4) involves only the element of area dA and not any details of the shape of the isotherms, the result (5) is also independent of the geometry of the flow. The same result (5) would apply to a onedimensional channel flow situation, rather than the irregular, roughly circular geometry sketched in Fig. 3a, provided the horizontal area A available for double-diffusive convection was the same. The parameter C, defined in (5), is the nondimensional measure of the relative importance of double-diffusive convection and advection. When C is large double diffusion is important. Equation (5) is plotted as the solid line in Fig. 4. We now impose that 0 e, the value of 0 at the centre of the Greenland Gyre, is 0.85 and find that the model gives C = 2.646. Using ( T A - T s ) = 4.1°C, ct = 0.5 x 10-4 K -t, A = 183,200 km 2 (from CARMACKand AAGAARD, 1973, Table 3), and R p = l . 6 , we obtain a volume flow rate Q0 of 3 x 10 ~m 3s -1 (95,500 km 3 y-t). Such a production rate of Bottom Water has been calculated using the winter temperature data, and we expect that the volume produced on an annual basis will be substantially <95,500 km 3 y-t. Carmack and Aagaard's estimate of 30,000 km 3 y-t is just less than a third of our winter flow rate, so this seems to be in good agreement given the assumptions of our model. A TEST OF THE IMPORTANCE OF CABBELING AT A HORIZONTAL INTERFACE
By estimating the biological consumption rate of oxygen in the Deep Water in the Greenland Sea, Carmaek and Aagaard were able to show that <5% of the Bottom Water production rate could be attributed to the highly oxygenated surface layer. We now use their observation to determine an upper limit to the relative importance of cabbeling and doublediffusive convection at the horizontal interface separating surface water from modified Atlantic Water. The relative importance of the nonlinear equation of state (or of cabbeling) and doublediffusive convection is conveniently expressed by the entrainment parameter /~, defined in terms of the heat fluxes as
~=
u e(T- Ts) 2F r + ue(T - Ts )
(6)
I 114
TREVORJ. McDOUGALL
(McDOUGALL, 19818), where u e is the entrainment velocity of fluid entering the modified Atlantic Water flow from the surface layer. If cabbeling were the only process at work then Z~ = I, whereas if double diffusion is dominant, L" ~ 0.0. We assume that u e ( T - Ts) is some constant fraction e o f F r [so that ~7 = 8/(2 + 5)] at all positions in the gyre and we seek to determine the value of 8 that gives the total entrained volume of surface water (fu~dA) equal to 5'?/,) of the total flow rate. The heat conservation equation,
QdT = -FrdA
-
tte(T-
Ts)dA = - ( I + 8)FrdA
fT)
now involves advection, double diffusion, and entrainment. The volume flow rate Q is a function of horizontal position and we define q = Q/ Qo. Noting that ( T - T s) = ( T A - Ts) (1- 0), the definition of the entrainment velocity u e ( T - T s ) = e f T leads to dq =
u,dA
Qo
eF T
=
(TA- T s)(1-O)
e
dA -
q
(1 +8)
dO
(8)
(1-0)'
where we have used (7) to change the variable of integration from A to 0. Upon integrating (8) we obtain
q~
O = ( 1 - 0 ) -~'+~, Q0
(9)
and using this in (7) we derive the following integral relationship for 0c as a function of C 0c 1 7 : - - - - - (1 - O)--l-~/l+~ dO. -10 t i+ 8) t*
C= [
(lO)
A 5% contribution from the surface layer by entrainment means that qc --- (0.95) -1 and with 0 c = 0.85 we obtain 6 = 0.028 from (9). This means that an entrained volume flux of 5% of the total volume flow rate is consistent with the flux of heat due to entrainment being 2.8% of the double-diffusive heat flux (or 2.7% of the total heat flux). The entrainment parameter ~? is 0.014 and is consistent with the low values of E measured in recent laboratory experiments [see McDOUGALL (1981b) Fig. 6 with Rp = 1.6 and a cabbeling parameter 8 = 1]. Performing the integral (10) with s = 0.028 we find that for 0e = 0.85, C is increased over its value with 8 = 0 by only 0.1%, an amount of entrainment that makes an insignificant change to the volume flow rate Q0 of Atlantic Water involved in the production of Bottom Water. T H E E F F E C T OF E N T R A I N M E N T OF B O T T O M W A T E R I N T O TH E C O N V E R G I N G SUBSURFACE LAYER
We use a simple model of entrainment from below into the modified Atlantic Water flow in an attempt to determine the sensitivity of our solution C = C(0 c) to such entrainment. We assume that the entrained buoyancy flux of Bottom Water into the modified Atlantic Water is a constant fraction k of the buoyancy flux across the upper surface o f the flowing layer. The buoyancy difference between the modified Atlantic Water and Bottom Water is ga(TA - Ts)(Oc - 0)(1 - Rf)/po, the entrainment velocity ue is given by U~a(TA -
rs)(O ~ -
O) = k a y r,
(1 l)
Greenland Sea Bottom Water formation
and k is taken from laboratory experiments to be 0.2 heat now takes the form (cf. 7)
1 1 15
(KANTHA,1980). The
conservation of
Q d T = -(I + k)FTdA,
(I 2)
and from the two equations we obtain q=
I- ~ c ) -WI+~
(13)
and C=
I +k)
1-
(1 - 0)4 dO.
(14)
As the centre of the gyre is approached 0 ~ 0c and the volume flow rate tends to infinity. This occurs because the energy argument (I l) over-estimates the entrainment velocity for small values of the interracialRichardson number defined by g6ph/PoU2,, where u, is die convective velocity scale (u3,= buoyancy flux × h), h is the depth of the flowing layer, and gSP/P0 is the buoyancy difference between the modified Atlantic Water and Bottom Water. By using values for the parameters appropriate to the Greenland Sea we fred that a Richardson number of unity occurs close to the centre of the gyre where 0 = 0 c - 0.0005 and q = 3.45. At that point the flow consists of Q0 of water from the Atlantic and 2.45Q 0 of entrained (recycled) Bottom Water. Even though q -~oo as 0 ~ 0c, the integral (14) for C(0c) is well-behaved as 0 ~ 0~. It can be shown that the small central part of the flow where the intarfacial Richardson number is < I contributes < I% of the value of the integral for C in (14). The integral relationship C(0¢) was
i.o
0,8
i
~.~. ~
~" ~ ~
~ ' a
0.6
0c 0.4
02
0
I
!
!
,.0
20
3.0
4D
C
Fig. 4. The nondimensionai temperature at the c~ntr¢ of the gyr¢, 0e, as a function of the parameter C, which r¢presen~ the relative ~ml~rt~ce of d o u b M - d ~ ."re convection and advection. The full line is the model without entrainment and the dashed line includes the entrainment of Bottom Water from below into the flowing modified Atlantic Water layer.
1116
TREVOR
J. MCDOUGALL
integrated numerically fork= 0.2 (Fig. 4). At ec = 0.85, the value of Cis approximately 13% larger than the value without entrainment. Discounting the 1% due to the unphysically large entrainment velocities at the very centre of the converging flow, we arrive at a figure of just 12% for the increase in the value of C caused by entrainment, even though the flow rate increases by a factor of 3.45. The insensitivity of C to entrainment is because such entrainment has two competing effects on the evolution of temperature as a function of area. The entrainment of Bottom Water makes the flow cooler and so increases 9, but the larger volume flow rate also reduces the rate of change of 9 with area (see 12). The two effects almost cancel so that the function C(9c) (where Cis defined using the initial volume flow rate Q0 ) is not very sensitive to even large amounts of entrained flow. A value of C that is 12% larger due to the inclusion of entrainment means that the required Qo to achieve ec = 0.85 is reduced by 12% from 95,500 to 84,000 km 3 y-l, 2.8 times the estimated renewal rate of Bottom Water between summer and late winter of 30,000 km 3 y- 1• The value of (TA - Ts) we have used is that appropriate to winter conditions, and because of the seasonally cyclic nature of Bottom Water formation, our estimate is certainly in better agreement with the observations than the factor of 2.8 would suggest. DISCUSSION
We have presented an idealized model of the production of Bottom Water in the Greenland Sea based on the observations of CARMACK and AAGAARD (1973). Double-diffusive convection between the surface layer and the subsurface Atlantic Water tongue makes the flow cooler, less saline, and denser. The model gives a value for the volume flow rate Q0 of inflowing Atlantic Water consistent with such water reaching the density of Bottom Water at the centre of the gyre. A larger flow rate would mean that double diffusion was not sufficiently strong to cause the Bottom Water density to be reached, while a smaller rate would mean that the modified Atlantic Water would sink before it reached the centre of the gyre, which is not observed. The production rate of Greenland Sea Bottom Water is not accurately known, but Carmack and Aagaard estimated the differences in the volume of Bottom Water in the Greenland Gyre between winter and summer, and our model gives encouraging agreement with this annual volume. An important assumption of the model is that of a constant density anomaly ratio R P across the double-diffusive interface. The assumption was suggested by Carmack and Aagaard's observations and it simplifies the analysis considerably, but variations in the value of RP will affect the double-diffusive fluxes as R~2 • We note that with RP > 1 there is a significant density difference between the surface water and the modified Atlantic Water, even at the centre of the gyre, consistent with there being only a 5% contribution of surface water to the Bottom Water production rate. The oxygen budget in the Greenland Sea has confirmed the laboratory results of McDouGALL (1981b) that cabbeling is relatively unimportant in comparison with doublediffusive <;onvection at a horizontal interface between two water masses. Figure 2 shows that the line from the Atlantic water point A to the Bottom Water point BW is almost tangent to the isopycnal through BW and so the conventional criterion for cabbeling between the two water masses is fulfilled. We do not, however, believe cabbeling to be important here for three reasons. Firstly, the winter temperature section across the Greenland Sea of Carmack and Aagaard (their Fig. 12a) shows an intruding tongue of Atlantic Water that does not sink until it reaches the centre of the gyre. If a significant amount of cabbeling were occurring, the
Greenland Sea Bottom Water formation
1117
isotherms would be expected to move to deep levels in the gyre as the mixed cabbeling water sank. Secondly, by analogy with the results of McDoUGALL(198 lb), static stability would be expected to inhibit the vertical mixing that is a prerequisite for cabbeling in this situation, even though there is no sharp interface between the two water masses. Thirdly, although cabbeling can in general form denser water than either contributing water mass, it cannot change the temperature and salinity except by mixing between the parent water masses, that is, on a straight line on the T - S diagram. On the other hand, the formation of Bottom Water from Atlantic Water requires the temperature and salinity of the water to change in a proportion consistent with double-diffusive convection between the surface water and the modified Atlantic Water. An important assumption in the model is that of a steady state. Carmack and Aagaard give the total volume of Atlantic Water in the Greenland Basin as 167,000 km 3, several times the annual production of Bottom Water. Perhaps with more detailed knowledge of seasonal processes in the Greenland Sea it may be worth considering a time-dependent model, but until then we believe that the present model captures the important physical balance between advection and double-diffusive convection. We have studied the processes that change the temperature, salinity, and density of the subsurface Atlantic Water layer as it converges to the centre of the Greenland Gyre. Here the density of the underlying Bottom Water is reached and sinking occurs. While we have concentrated on the changes to the subsurface layer, the cooling of the surface layer by the atmosphere is important in the model as it keeps the surface layer temperature at the freezing point. We consider the model to be complementary to the work of KaLLWORTH(1979), who suggested that narrow sinking 'chimneys' form by baroclinic instability in such an area as the central Greenland Sea. The subsurface modification of Atlantic Water by double diffusion in our model provides the necessary changes in temperature and salinity to achieve those of Bottom Water, whereas the chimney theory of KILLWORTH'(1979)explains why the sinking regions of freshly produced Bottom Water are so small in horizontal extent and consequently so difficult to observe. REFERENCES
CARMACK E. C. (1979) Combined influence of inflow and lake temperatures on spring circulation in a riverine lake. Journal of Physical Oceanography, 9, 422-434. CARMACK E. and K. AAGAARD (1973) On the Deep Water of the Greenland sea. Deep-Sea Research, 20. 687-715. HUPPERT H. E. (1971) On the stability of a series of double-diffusive layers. Deep-Sea Research, 18, 1005-102 I. KANTHA L. H. (1980) Turbulent entrainment at a buoyancy interface due to convective turbulence. In: Fjord oceanography, H.J. FREELAND, D. M. FARMERand C. D. LEVlNGS, editors, Plenum Press, New York, pp. 205-2 i 3. KILLWORTH P. D. (19"/9) On "chimney" formations in the ocean. Journal of Physical Oceanography, 9, 531-554. MAMAYEV O, 1. (1975) Temperature-salinity analysis of world ocean waters. English Edition, Elsevier. Amsterdam, 3"/4 pp. McDOUGALL T. J. (198 In) Double-diffusive convection with a non-linear equation of state. 1. The accurate conservation of properties in a two-layer system. Progress in Oceanography, 10, 71-89. M cDOUGALL T. J. (1981 b) Double-diffusive convection with a non-linear equation of state. 11. Laboratory experiments and their interpretation. Progress In Oceanography, 10, 9 !-121. TURNER J. S. (1965) The coupled turbulent transports of salt and heat across a sharp density interface. International Journal of Heat and Mass Transfer, 8, 759-767.
0198-0149/83 $3.00 + 0.00 O 1983Pergamon Press Ltd.
Greenland Sea Bottom Water formation: a balance between adveetion and double-diffusion TREVORJ. McDOUGALL* (Received 31 January 1983; accepted 5 April 1983; final revision received 19April 1983) Abstract--Greenland Sea Bottom Water is believed to form by the subsurfzce modification of Atlantic Water (CARMACK and AAOAARD, 1973, Deep-Sea Research, 20, 687-715) rather than by direct cooling at the surface. Both double-diffusive convection and cabbeling have been suggested as responsible for the increase in density of the Atlantic Water, but we argue against the importance of cabbeling. A simple model is presented in which there is a balance between advection and double diffusion; it is applied to the Atlantic Water as it flows horizontally towards the centre of the Greenland Gyre. The model gives a prediction for the rate of formation of Bottom Water that is encouragingly close to that observed.
INTRODUCTION
CARMACK and AAGAARD(1973) conducted a volumetric census of water masses in the Greenland Sea for both summer and winter and found that the volume of Bottom Water (temperature < - I ° C ) varied seasonally by (30 + 10) x 103 km 3. This is a lower limit to the annual production of Bottom Water in the Greenland Sea because the actual production rate must also include the volume of Bottom Water that flows out of the Greenland Sea from one summer to the following winter. The budget of dissolved oxygen rules out the surface mixed layer as the major source of Bottom Water and suggests that Atlantic Water, which flows into the Greenland Sea just below the surface layer, is first modified by double-diffusive convection and then sinks as Bottom Water near the centre of the gyre. Figure 1 (from CARMACKand AAOAARD, 1973) shows the temperatures and salinities of both surface water and the subsurface, modified Atlantic Water at several locations in the Greenland Sea. The surface measurements are joined by straight lines to the corresponding subsurface points at the same location. As the centre of the gyre is approached, the salinity of the surface layer increases and the temperature and salinity of the modified Atlantic Water decrease. The temperature of the surface layer remains almost constant at the freezing point. Figure 2 (from CARMACKand AAGAARD, 1973) shows the inflowing Atlantic Water (point A) and the Bottom Water (point BW) on a temperature-salinity (T-S) diagram. Double-diffusive convection (in the diffusive sense) between the well-mixed surface layer and the Atlantic Water causes the temperatures and salinities of the water masses to move in the directions shown by the arrows in Fig. 2. The arrows have been drawn parallel to the line joining points A and BW, and the slope of the line corresponds to a buoyancy flux ratio R/of0.56 Itaking ct=0.5 × 10-4 k -I and 13=0.765 × 10-3 (%0)-I from MAMAYEV, 19751. The density * Research School of Earth Sciences, Australian National University, P.O. Box 4, Canberra 2600, Australia. Present address: Division of Oceanography, C.S.LR.O., P.O. Box 1538, Hobart, Tasmania 7001, Australia. 1109
1110
TREVOR J. McDOUGALL
2f~
...................................................................
I / /£ SA
' 3
~_~ ;o n
-
3400
34.20
34.40
34.60
34.80
35.00
35.20
SaLinity (xl03)
Fig. 1. Temperature-pl_init¥ diagram showing surface observations (O)joined to the corresponding points (O) of the modified Atilnti¢ Water layer. The line SA is tangent at the mean temperature - salinity point of Bottom Water BW, to the isopyenal passing.through the point. [From CARMACX and AAOAARD (1973), Fig. 5.]
anomaly ratio, Rp is 1.6 for the two water masses and so while the buoyancy flux ratio is consistent with the measurements by TURNER (1965), it is a little higher than those of
McDoUOALL (198 lb). The line in Fig. 2 connecting the surface water S to the Atlantic Water A is tangent to the isopycnal through point A. This is the conventional criterion for the onset of cabbeling where mixtures of two water masses are more dense than either parent water mass. When two water masses lie one above the other and the criterion for cabbeling is fulfilled, the vertical distributions of temperature and salinity are also in the correct sense for the occurrence of doublediffusive convection. McDOUOALL (1981b) investigated the relative importance of the two
~) 2
E o S'
-I
34.40
34.60
BW
34.80
3500
35.20
Satinity ( x l 0 3)
Fig. 2. Temperature-salinity diagram showing the original (i.e., at the edge of the gyre) Atlantic Water A. the original Surface Polar Water S, and the Bottom Wat~ BW. Double-diffusive fluxes of heat and salt alter A in the direction A' and S in the direction S'. The surface layer also experiences cooling from the atmosphere, which is not shown on the figure. [From CARMACK and AAGAARD (1973), Fig. 7.]
Greenland Sea Bottom Water formation
1111
processes in such geometry and showed that the prime effects of the nonlinearity in the equation of state are firstly to drive a greater level of double-diffusive convective activity in the lower layer than in the upper layer, and secondly, to make the lower edge of the interfacial region less gravitationally stable. Both effects cause the horizontal interface between the water masses to migrate slowly upwards as the lower layer entrains fluid from the upper layer. While cabbeling is effective in causing deep convection by mixing between horizontally adjacent water masses (CARMACK, 1979), McDOUOALL(1981b) showed that when such potentially cabbeling water masses lie one above the other, the vertical static stability of the water column inhibits any large-scale sinking due to cabbeling, and double-diffusive convection is the dominant process. In this paper we seek to understand the processes that cause the properties of Atlantic Water to change as it flows towards the centre of the Greenland Gyre. Firstly we consider a simple model, which is a balance between (a) double-diffusive convection across the horizontal interface and (b) the horizontal advection of the modified Atlantic Water. When the fluid reaches the centre of the gyre it is assumed to have reached the density of.the water below it and then to sink as Bottom Water. The assumption has the nature of a consistency relationship and it gives a production rate of Bottom Water close to that observed. The model is then extended to include firstly the entrainment of surface layer fluid into the modified Atlantic Water. This, together with the oxygen measurements by Carmaek and Aagaard, confirm the conclusions of McDOUGALL(198 l b) that such entrainment is small even though the nonlinearity of the equation of state is significant. Finally, we include the entrainment of Bottom Water into the converging Atlantic Water flow and show that this can at most change our estimate of Bottom Water production rate by 12%. THE B A L A N C E B E T W E E N A D V E C T I O N A N D D O U B L E - D I F F U S I V E C O N V E C T I O N
The basic features of the model are illustrated in Fig. 3. Atlantic Water converges at flow rate Q0 to the central sink in the Greenland Gyre, continually losing heat and salt double diffusively to the surface mixed layer. The fluxes change the temperature and salinity of the converging layer and when the centre of the gyre is reached the fluid is assumed to have the same properties as the underlying Bottom Water, whereupon it sinks. This assumption enables the rate of production of bottom water Q0 to be estimated. If the volume flow rate were > Q0, then double-diffusive convection would not be strong enough to change the layer atmosphere
o°'r:'
I"
"
"
AW
AW
BW
t new
m i
BW
BW
T
T+dT
(a)
o.
(b) Fig. 3. The idealizedmodelin plan view(a) and in cross-section(b). In (a), dA is an elementof the horizontal area between the isotherms T and T + dT. In (b) Atlantic Water (AW) enters at a flow rate Q0 between the surface mixed layer and the Bottom Water (BW).
1 112
I'REVORJ. McDOUGALL
properties to those of Bottom Water and so sinking and the production of Bottom Water would not occur. If, on the other hand, the volume flow rate were
(l)
where a is the thermal expansion coefficient of seawater (defined so a > 0), P0 is a reference density, Cp is the specific heat (at constant pressure) of seawater, and b is a dimensional constant (dimensions of velocity) = 0.323 (g~/v)t, where the symbols have their usualmeaning. For water at room temperature b can be taken as 1.9 × 10 -3 m s -1. AT is the temperature difference across the interface and Rp is the density anomaly ratio, ~ A S / a A T . Consider now the conservation of heat for a control volume of the Atlantic Water layer between the isotherms T and T + dT in Fig. 3a. The horizontal area of the control volume is dA and the depth is the depth of the Atlantic Water layer. The steady-state conservation of heat involves only the terms due to advection and double-diffusive convection and is given by QoadT = - a F r d A .
(2)
The negative sign arises because the modified Atlantic Water is cooled by the double diffusion. Neither the depth of the flowing layer nor the velocity of the flow appear separately in (2), but only the product of these variables multiplied by the circumference of the control volume is relevant and is labelled Q0. We note (as did Carmack and Aagaard) from Fig. 1 that the lines on the T - S diagram joining the successive surface water points to the modified Atlantic Water points have a remarkably constant slope that corresponds to a density anomaly ratio R p of 1.6. The temperature of the surface layer is kept virtually constant at the freezing point by exchange of heat with the atmosphere, and the surface salinity increases due to the double-diffusive flux F s from below. The surface layer also undergoes an unknown amount of adveetion and other processes. Without endeavouring to understand all the processes at work on the surface layer, we make use in our model of the observed constancy of both the surface layer temperature T~ and the density anomaly ratio Rp. We 'define a nondimensional temperature scale by TA-T
0= - -
(3)
TA - T s
where TA is the initial temperature of the inflowing Atlantic Water and T is the temperature of the flow as a function of horizontal position. The temperature and salinity of the modified
Greenland Sea Bottom Water formation
1113
Atlantic Water layer proceed from point A in Fig. 2 towards point BW. The nondimensional temperature 0 varies from zero at point A to 1 when (T, S) is at the point on the extension of the line A to BW where T = T s . From Fig. 1, 0 -~ 0.85 when the temperature and salinity of the modified Atlantic Water are those of Bottom Water (BW). Combining (1), (2), and (3), the advective, double-diffusive balance is dO =
bl(~(T A - Ts)]~
QoR20
(1 - 0)] dA.
(4)
The terms in 0 are readily separated to the left-hand side of the equation and we integrate over the total horizontal area A from 0 = 0 at the outside to 0 = 0 c at the centre of the gyre, giving C -
b [ a ( T A - Ts)]~A
a0R
= f ~ ( 1 - 0)-~d0 = 3(1 - 0~)-~ - 3.
(5)
Because the advective, double-diffusive balance given by (2) or (4) involves only the element of area dA and not any details of the shape of the isotherms, the result (5) is also independent of the geometry of the flow. The same result (5) would apply to a onedimensional channel flow situation, rather than the irregular, roughly circular geometry sketched in Fig. 3a, provided the horizontal area A available for double-diffusive convection was the same. The parameter C, defined in (5), is the nondimensional measure of the relative importance of double-diffusive convection and advection. When C is large double diffusion is important. Equation (5) is plotted as the solid line in Fig. 4. We now impose that 0 e, the value of 0 at the centre of the Greenland Gyre, is 0.85 and find that the model gives C = 2.646. Using ( T A - T s ) = 4.1°C, ct = 0.5 x 10-4 K -t, A = 183,200 km 2 (from CARMACKand AAGAARD, 1973, Table 3), and R p = l . 6 , we obtain a volume flow rate Q0 of 3 x 10 ~m 3s -1 (95,500 km 3 y-t). Such a production rate of Bottom Water has been calculated using the winter temperature data, and we expect that the volume produced on an annual basis will be substantially <95,500 km 3 y-t. Carmack and Aagaard's estimate of 30,000 km 3 y-t is just less than a third of our winter flow rate, so this seems to be in good agreement given the assumptions of our model. A TEST OF THE IMPORTANCE OF CABBELING AT A HORIZONTAL INTERFACE
By estimating the biological consumption rate of oxygen in the Deep Water in the Greenland Sea, Carmaek and Aagaard were able to show that <5% of the Bottom Water production rate could be attributed to the highly oxygenated surface layer. We now use their observation to determine an upper limit to the relative importance of cabbeling and doublediffusive convection at the horizontal interface separating surface water from modified Atlantic Water. The relative importance of the nonlinear equation of state (or of cabbeling) and doublediffusive convection is conveniently expressed by the entrainment parameter /~, defined in terms of the heat fluxes as
~=
u e(T- Ts) 2F r + ue(T - Ts )
(6)
I 114
TREVORJ. McDOUGALL
(McDOUGALL, 19818), where u e is the entrainment velocity of fluid entering the modified Atlantic Water flow from the surface layer. If cabbeling were the only process at work then Z~ = I, whereas if double diffusion is dominant, L" ~ 0.0. We assume that u e ( T - Ts) is some constant fraction e o f F r [so that ~7 = 8/(2 + 5)] at all positions in the gyre and we seek to determine the value of 8 that gives the total entrained volume of surface water (fu~dA) equal to 5'?/,) of the total flow rate. The heat conservation equation,
QdT = -FrdA
-
tte(T-
Ts)dA = - ( I + 8)FrdA
fT)
now involves advection, double diffusion, and entrainment. The volume flow rate Q is a function of horizontal position and we define q = Q/ Qo. Noting that ( T - T s) = ( T A - Ts) (1- 0), the definition of the entrainment velocity u e ( T - T s ) = e f T leads to dq =
u,dA
Qo
eF T
=
(TA- T s)(1-O)
e
dA -
q
(1 +8)
dO
(8)
(1-0)'
where we have used (7) to change the variable of integration from A to 0. Upon integrating (8) we obtain
q~
O = ( 1 - 0 ) -~'+~, Q0
(9)
and using this in (7) we derive the following integral relationship for 0c as a function of C 0c 1 7 : - - - - - (1 - O)--l-~/l+~ dO. -10 t i+ 8) t*
C= [
(lO)
A 5% contribution from the surface layer by entrainment means that qc --- (0.95) -1 and with 0 c = 0.85 we obtain 6 = 0.028 from (9). This means that an entrained volume flux of 5% of the total volume flow rate is consistent with the flux of heat due to entrainment being 2.8% of the double-diffusive heat flux (or 2.7% of the total heat flux). The entrainment parameter ~? is 0.014 and is consistent with the low values of E measured in recent laboratory experiments [see McDOUGALL (1981b) Fig. 6 with Rp = 1.6 and a cabbeling parameter 8 = 1]. Performing the integral (10) with s = 0.028 we find that for 0e = 0.85, C is increased over its value with 8 = 0 by only 0.1%, an amount of entrainment that makes an insignificant change to the volume flow rate Q0 of Atlantic Water involved in the production of Bottom Water. T H E E F F E C T OF E N T R A I N M E N T OF B O T T O M W A T E R I N T O TH E C O N V E R G I N G SUBSURFACE LAYER
We use a simple model of entrainment from below into the modified Atlantic Water flow in an attempt to determine the sensitivity of our solution C = C(0 c) to such entrainment. We assume that the entrained buoyancy flux of Bottom Water into the modified Atlantic Water is a constant fraction k of the buoyancy flux across the upper surface o f the flowing layer. The buoyancy difference between the modified Atlantic Water and Bottom Water is ga(TA - Ts)(Oc - 0)(1 - Rf)/po, the entrainment velocity ue is given by U~a(TA -
rs)(O ~ -
O) = k a y r,
(1 l)
Greenland Sea Bottom Water formation
and k is taken from laboratory experiments to be 0.2 heat now takes the form (cf. 7)
1 1 15
(KANTHA,1980). The
conservation of
Q d T = -(I + k)FTdA,
(I 2)
and from the two equations we obtain q=
I- ~ c ) -WI+~
(13)
and C=
I +k)
1-
(1 - 0)4 dO.
(14)
As the centre of the gyre is approached 0 ~ 0c and the volume flow rate tends to infinity. This occurs because the energy argument (I l) over-estimates the entrainment velocity for small values of the interracialRichardson number defined by g6ph/PoU2,, where u, is die convective velocity scale (u3,= buoyancy flux × h), h is the depth of the flowing layer, and gSP/P0 is the buoyancy difference between the modified Atlantic Water and Bottom Water. By using values for the parameters appropriate to the Greenland Sea we fred that a Richardson number of unity occurs close to the centre of the gyre where 0 = 0 c - 0.0005 and q = 3.45. At that point the flow consists of Q0 of water from the Atlantic and 2.45Q 0 of entrained (recycled) Bottom Water. Even though q -~oo as 0 ~ 0c, the integral (14) for C(0c) is well-behaved as 0 ~ 0~. It can be shown that the small central part of the flow where the intarfacial Richardson number is < I contributes < I% of the value of the integral for C in (14). The integral relationship C(0¢) was
i.o
0,8
i
~.~. ~
~" ~ ~
~ ' a
0.6
0c 0.4
02
0
I
!
!
,.0
20
3.0
4D
C
Fig. 4. The nondimensionai temperature at the c~ntr¢ of the gyr¢, 0e, as a function of the parameter C, which r¢presen~ the relative ~ml~rt~ce of d o u b M - d ~ ."re convection and advection. The full line is the model without entrainment and the dashed line includes the entrainment of Bottom Water from below into the flowing modified Atlantic Water layer.
1116
TREVOR
J. MCDOUGALL
integrated numerically fork= 0.2 (Fig. 4). At ec = 0.85, the value of Cis approximately 13% larger than the value without entrainment. Discounting the 1% due to the unphysically large entrainment velocities at the very centre of the converging flow, we arrive at a figure of just 12% for the increase in the value of C caused by entrainment, even though the flow rate increases by a factor of 3.45. The insensitivity of C to entrainment is because such entrainment has two competing effects on the evolution of temperature as a function of area. The entrainment of Bottom Water makes the flow cooler and so increases 9, but the larger volume flow rate also reduces the rate of change of 9 with area (see 12). The two effects almost cancel so that the function C(9c) (where Cis defined using the initial volume flow rate Q0 ) is not very sensitive to even large amounts of entrained flow. A value of C that is 12% larger due to the inclusion of entrainment means that the required Qo to achieve ec = 0.85 is reduced by 12% from 95,500 to 84,000 km 3 y-l, 2.8 times the estimated renewal rate of Bottom Water between summer and late winter of 30,000 km 3 y- 1• The value of (TA - Ts) we have used is that appropriate to winter conditions, and because of the seasonally cyclic nature of Bottom Water formation, our estimate is certainly in better agreement with the observations than the factor of 2.8 would suggest. DISCUSSION
We have presented an idealized model of the production of Bottom Water in the Greenland Sea based on the observations of CARMACK and AAGAARD (1973). Double-diffusive convection between the surface layer and the subsurface Atlantic Water tongue makes the flow cooler, less saline, and denser. The model gives a value for the volume flow rate Q0 of inflowing Atlantic Water consistent with such water reaching the density of Bottom Water at the centre of the gyre. A larger flow rate would mean that double diffusion was not sufficiently strong to cause the Bottom Water density to be reached, while a smaller rate would mean that the modified Atlantic Water would sink before it reached the centre of the gyre, which is not observed. The production rate of Greenland Sea Bottom Water is not accurately known, but Carmack and Aagaard estimated the differences in the volume of Bottom Water in the Greenland Gyre between winter and summer, and our model gives encouraging agreement with this annual volume. An important assumption of the model is that of a constant density anomaly ratio R P across the double-diffusive interface. The assumption was suggested by Carmack and Aagaard's observations and it simplifies the analysis considerably, but variations in the value of RP will affect the double-diffusive fluxes as R~2 • We note that with RP > 1 there is a significant density difference between the surface water and the modified Atlantic Water, even at the centre of the gyre, consistent with there being only a 5% contribution of surface water to the Bottom Water production rate. The oxygen budget in the Greenland Sea has confirmed the laboratory results of McDouGALL (1981b) that cabbeling is relatively unimportant in comparison with doublediffusive <;onvection at a horizontal interface between two water masses. Figure 2 shows that the line from the Atlantic water point A to the Bottom Water point BW is almost tangent to the isopycnal through BW and so the conventional criterion for cabbeling between the two water masses is fulfilled. We do not, however, believe cabbeling to be important here for three reasons. Firstly, the winter temperature section across the Greenland Sea of Carmack and Aagaard (their Fig. 12a) shows an intruding tongue of Atlantic Water that does not sink until it reaches the centre of the gyre. If a significant amount of cabbeling were occurring, the
Greenland Sea Bottom Water formation
1117
isotherms would be expected to move to deep levels in the gyre as the mixed cabbeling water sank. Secondly, by analogy with the results of McDoUGALL(198 lb), static stability would be expected to inhibit the vertical mixing that is a prerequisite for cabbeling in this situation, even though there is no sharp interface between the two water masses. Thirdly, although cabbeling can in general form denser water than either contributing water mass, it cannot change the temperature and salinity except by mixing between the parent water masses, that is, on a straight line on the T - S diagram. On the other hand, the formation of Bottom Water from Atlantic Water requires the temperature and salinity of the water to change in a proportion consistent with double-diffusive convection between the surface water and the modified Atlantic Water. An important assumption in the model is that of a steady state. Carmack and Aagaard give the total volume of Atlantic Water in the Greenland Basin as 167,000 km 3, several times the annual production of Bottom Water. Perhaps with more detailed knowledge of seasonal processes in the Greenland Sea it may be worth considering a time-dependent model, but until then we believe that the present model captures the important physical balance between advection and double-diffusive convection. We have studied the processes that change the temperature, salinity, and density of the subsurface Atlantic Water layer as it converges to the centre of the Greenland Gyre. Here the density of the underlying Bottom Water is reached and sinking occurs. While we have concentrated on the changes to the subsurface layer, the cooling of the surface layer by the atmosphere is important in the model as it keeps the surface layer temperature at the freezing point. We consider the model to be complementary to the work of KaLLWORTH(1979), who suggested that narrow sinking 'chimneys' form by baroclinic instability in such an area as the central Greenland Sea. The subsurface modification of Atlantic Water by double diffusion in our model provides the necessary changes in temperature and salinity to achieve those of Bottom Water, whereas the chimney theory of KILLWORTH'(1979)explains why the sinking regions of freshly produced Bottom Water are so small in horizontal extent and consequently so difficult to observe. REFERENCES
CARMACK E. C. (1979) Combined influence of inflow and lake temperatures on spring circulation in a riverine lake. Journal of Physical Oceanography, 9, 422-434. CARMACK E. and K. AAGAARD (1973) On the Deep Water of the Greenland sea. Deep-Sea Research, 20. 687-715. HUPPERT H. E. (1971) On the stability of a series of double-diffusive layers. Deep-Sea Research, 18, 1005-102 I. KANTHA L. H. (1980) Turbulent entrainment at a buoyancy interface due to convective turbulence. In: Fjord oceanography, H.J. FREELAND, D. M. FARMERand C. D. LEVlNGS, editors, Plenum Press, New York, pp. 205-2 i 3. KILLWORTH P. D. (19"/9) On "chimney" formations in the ocean. Journal of Physical Oceanography, 9, 531-554. MAMAYEV O, 1. (1975) Temperature-salinity analysis of world ocean waters. English Edition, Elsevier. Amsterdam, 3"/4 pp. McDOUGALL T. J. (198 In) Double-diffusive convection with a non-linear equation of state. 1. The accurate conservation of properties in a two-layer system. Progress in Oceanography, 10, 71-89. M cDOUGALL T. J. (1981 b) Double-diffusive convection with a non-linear equation of state. 11. Laboratory experiments and their interpretation. Progress In Oceanography, 10, 9 !-121. TURNER J. S. (1965) The coupled turbulent transports of salt and heat across a sharp density interface. International Journal of Heat and Mass Transfer, 8, 759-767.